Automatic analysis of computed catalytic cycles


  • Andreas Uhe,

    Corresponding author
    1. Institut für Technische und Makromolekulare Chemie, RWTH Aachen University, Worringerweg 1, Aachen 52074, Germany
    • Institut für Technische und Makromolekulare Chemie, RWTH Aachen University, Worringerweg 1, Aachen 52074, Germany
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  • Sebastian Kozuch,

    Corresponding author
    1. Institute of Chemistry and the Lise Meitner-Minerva Center for Computational Quantum Chemistry, Hebrew University of Jerusalem, Givat Ram Campus, Jerusalem 91904, Israel
    Current affiliation:
    1. Department of Organic Chemistry, The Weizmann Institute of Science, IL-76100 Rehovot, Israel
    • Institut für Technische und Makromolekulare Chemie, RWTH Aachen University, Worringerweg 1, Aachen 52074, Germany
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  • Sason Shaik

    1. Institute of Chemistry and the Lise Meitner-Minerva Center for Computational Quantum Chemistry, Hebrew University of Jerusalem, Givat Ram Campus, Jerusalem 91904, Israel
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The energetic span model allows the estimation of the turnover frequency (TOF) of a catalytic reaction from its calculated energy profile. Furthermore, by identifying the TOF determining intermediate and the TOF determining transition state, the model shows that the concept of “determining states” is more useful and correct than the concept of “determining steps.” This article illustrates the application of the model and provides an introduction to its concepts using instructive examples. The first part explains the model in its current state of development, whereas in the second part the degree of TOF control of the reactant and product concentrations is introduced. With this information, it is possible to give explicit recommendations regarding the conditions to be applied in the experiment, e.g., which reactant promotes the reaction or if a product kinetically inhibits it. At the end, we present the AUTOF program that allows the user to apply the complete model in a black box fashion. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011


The turnover frequency (TOF) of a catalytic reaction is the traditional measure of the efficiency of a catalyst, expressed as the number of cycles performed per time unit and catalyst concentration. The TOF is routinely available in experimental chemistry, but until recently, there was no simple and straightforward method to calculate it from the conventional energy profiles obtained by means of computational chemistry techniques. Such a link is important as the interplay of theory and experiment can be extremely fruitful in catalysis.

The energetic span model developed by Kozuch and Shaik1–6 has the potential to interweave computational and experimental chemistry. It allows to calculate the TOF of a catalytic reaction from the energy profile4 by the following eq. (1), which gives the energy-representation (E-representation) of the TOF:

equation image(1)

Here ΔGr is the energy of the global reaction. Ti and Ij are the calculated free energies of each transition state and intermediate, respectively. δGij is either ΔGr or zero, according to the position of Tivs. Ij in the specific term of the summation, as specified in eq. (2):

equation image(2)

In eq. (1), Δ (the numerator of the TOF formulation) corresponds to the driving force of the reaction; whereas M is the kinetic resistance. In analogy to Ohm's law, a catalytical flux law can be established: TOF (the catalytic current) is equal to Δ (the catalytic potential) divided by M (the kinetic resistance to catalysis).3 The Δ and M terms were introduced by Christiansen7 to define the TOF as a function of kinetic rate constants (the k-representation)4 in a steady state regime. The Δ and M terms in the E-representation can be derived from Christiansen's formulation by applying Eyring's transition state theory.2

Using the E-representation in eqs. (1) and (2), the computational chemist can calculate TOFs and compare the efficiency of different catalytic systems from computational data, and therefore suggest the most suitable mechanism, catalyst, substrate, or solvent for experimental examination. Vice versa, experimentalists can test the computational study against their experimental TOF values, and thereby provide feedback to the computational chemist on the accuracy of his or her calculations.

Herein, we apply the energetic span model to a recent computational study regarding the hydroamination of ethylene with ammonia to form ethylamine, catalyzed by a rhodium pincer complex.8 In addition, we present the AUTOF program that allows the user to apply the complete model in a black box fashion.

Application of the Energetic Span Model: The Hydroamination Reaction

The direct catalytic formation of amines from nonactivated olefins and ammonia under mild conditions is a long standing goal of catalysis, which has not been achieved to date.9 Recently, a full catalytic cycle of this reaction was computed using a rhodium complex as catalyst and ethylene as model olefin.8 The active species I1 is a Rh NCN pincer complex (NCN = 2,5-bis (dimethylaminomethyl)-benzene) bearing an ethyl and an amide ligand (Fig. 1). The energy profile of the cycle is shown in Figure 2. The quantum mechanical (QM) calculations were produced at the RI-PBE-D/def2-TZVP level providing gas-phase free energies. All the kinetic information was obtained by applying the energetic span model using the AUTOF program, which is described later on.

Figure 1.

Optimized structure I1 seen (a) from top and (b) from the side. The hydrogen atoms of the pincer ligand were omitted for clarity.

Figure 2.

Free Energy profile for the catalytic hydroamination of ethylene with ammonia. The pincer ligand was omitted for clarity.

The catalytic cycle starts by coordination of ethylene (I2) followed by a migratory insertion into the Rh-N bond via T2 leading to I3. Breaking the coordination of the newly formed amine (I4) allows for the coordination of ammonia (I5). A σ-bond metathesis via T5 liberates ethylamine and regenerates the active species I1. Alternative pathways were considered in the original study, but turned out to be less favorable. They are neglected here as our focus is on the explanation of how to use and understand the energetic span model.

The example chosen has five steps (step 1: I1I2, step 2: I2I3, step 3: I3I4, step 4: I4I5, and step 5: I5I1), but only two transition states were calculated, namely, that of step 2 (T2) and that of step 5 (T5). The missing transition states were not determined as they correspond to low barrier association/dissociation steps. The influence of these missing transition states is negligible, but to apply the energetic span model one has to provide the energies of one intermediate and one transition state for each step of the catalytic cycle. Therefore, we assumed the missing transition states to have the same energy as the highest adjacent intermediate (for a more detailed explanation see the Supporting Information). This method leads to the values in Table 1 listing the intermediate and transition state energies for each step of the example reaction and the overall reaction energy ΔGr.

Table 1. Free Energy States for the Hydroamination Reaction of Figure 2
 Intermediates (kcal mol−1)Transition states (kcal mol−1)
  1. C2H4 + NH3 → C2H5NH2; ΔGr = −6.5.

Step 10.0I13.7T1
Step 23.7I218.9T2
Step 33.8I37.4T3
Step 47.4I47.4T4
Step 56.0I520.1T5

With the energies of Table 1, one has all the required state energies to calculate the TOF according to eq. (1). The numerator contains ΔGr, which is the thermodynamic information of the overall reaction. The denominator on the other hand contains the summation terms Mi,j that are calculated from all intermediate and transition state energies, which contain the kinetic information. As the Mi,j are exponential terms, their actual values can differ by several orders of magnitude and most of them can be neglected. Therefore, usually there is only one (or sometimes two) combination of a transition state (TS) and an intermediate that is actually significant for the TOF.

In the hydroamination example, the estimated TOF using eq. (1) is 530 h−1 at 323 K. The most influential Mi,j term corresponds to T5 and I1 (M5,1 = 9.95·1017). These states, being the most influential states, are called the TOF determining transition state (TDTS) and TOF determining intermediate (TDI). The combination of T2 and I1 provides a less important but still significant contribution (M2,1 = 1.53·1017). The rest of the Mi,j terms are completely negligible (see SI).

Knowing the TDI and TDTS is a big advantage for catalyst development. If one were to screen different catalysts or solvent effects, one would have to calculate only these determining species for each modification to decide if it has the potential to increase the TOF.5, 6 However, this approach must be used with care, as large modifications of the catalyst or its environment may change the mechanism, and a new potential surface estimation may then be needed.

The Degree of TOF Control and the Energetic Span Approximation

To identify the TDI and TDTS, the energetic span model offers a quantification of the influence of each intermediate and transition state on the TOF—the so-called “degree of TOF control” (XTOF).2 The XTOF was developed based on Campbell's degree of rate control10, 11 and measures how much will the TOF vary by a small change on a TS or intermediate energy. It is calculated according to eq. (3a) for intermediates and (3b) for transition states2:

equation image(3a)
equation image(3b)

Equation (3a) shows that the XTOF of an intermediate Ik is the sum of all the Mi,j terms that contain this intermediate energy divided by the sum of all Mi,j terms. Analogously the XTOF of a transition state Tk is the sum of all the Mi,j terms that contain this transition state energy divided by the sum of all Mi,j terms [eq. (3b)]. Table 2 shows the XTOF values for the hydroamination reaction.

Table 2. XTOF for the Hydroamination Reaction
 Xmath imageXmath image
Step 11.00I10.00T1
Step 20.00I20.13T2
Step 30.00I30.00T3
Step 40.00I40.00T4
Step 50.00I50.87T5

Again, we see that the only intermediate influencing the TOF is I1 (the TDI), whereas both transition states T2 and T5 are important with T5 being the most influential (the TDTS). A visual technique to find the TDI and TDTS can be found on appendix 2 of ref.3. Note that the sum of the Xmath image and the sum of the Xmath image are normalized to one.2, 3

Once the TDI and TDTS have been identified, a simple estimation of the TOF can be made by the energetic span approximation, which gives the model its name2:

equation image(4)


equation image(5)

δE, the energetic span, corresponds to the apparent activation energy of the full cycle. Thus, knowing the TDTS, the TDI, and when necessary also the reaction energy, this approximation provides all the relevant kinetic information that is required for calculating the TOF of the cycle. However, this approximation is only valid, if there is only one intermediate and one transition state with XTOF > 0.00. In the case of the hydroamination, this approximation leads to a TOF of 610 h−1 using I1 as TDI and T5 as TDTS. The deviation from the originally calculated TOF of 530 h−1 obtained with eq. (1) is caused by the neglected T2 state.

As can be seen in eqs. (4) and (5), the energy based formulation (E-representation) of the energetic span model leads to simple equations guiding the search for new and improved catalysts. The use of rate constants (k-representation) to describe the kinetics of the reaction leads to cumbersome equations involving products of many rate constants with less insight. By contrast, the E-representation provides a lucid physical picture showing the key factors of catalysis. Thus, in catalysis there are no rate determining steps, instead there are rate determining states.3, 6

The Influence of Spectator Species

To show the wide applicability of the energetic span model, we want to increase the complexity of the example reaction at this point. It was shown in the original study8 that the relatively stable intermediates I2b, I5b, and I5c are accessible without being part of the catalytic cycle as shown in Figure 3, i.e., these are spectator species. Such spectator intermediates are likely to decrease the TOF of a catalytic reaction, so the question arises, how this can be accounted for within the energetic span model?

Figure 3.

Free Energy profile for the catalytic hydroamination of ethylene with ammonia including spectator species I2b, I5b, and I5c. The pincer ligand was omitted for clarity.

The easiest way to include I2b, I5b, and I5c into the cycle is to increase the number of steps. One could assume that as I2b is more stable than I2 and I5c is more stable than I5 and I5b the reaction sequence is altered to I1I2bI2I3I4I5cI5bI5I1. The TOF that is predicted by the energetic span model using this sequence is 1.6 h−1 compared with the TOF of 530 h−1 that was calculated before, when the spectator species were not taken into account. So in line with the theory, stable spectator species may lower the TOF significantly. Therefore, they must be taken into account when applying the energetic span model. Table 3 shows the energies and XTOF values for the sequence above.

Table 3. Free Energies and Degrees of TOF Control (XTOF Values) for the Hydroamination Including Spectator Species, as Appears in Figure 3
Energy (kcal mol−1)XTOF
IntermediatesTransition statesIntermediatesTransition states

According to Table 3, not only the TOF but also the XTOF values change dramatically, when the spectator species are taken into account. This leads to a completely different picture of the important species in this catalytic system. First, intermediate I1 is not TOF determining anymore, as the low lying species I2b, I5b, and I5c will draw most of the catalyst concentration. Among these three, I5c is by far the most important as it has the highest XTOF. Second, in this mechanism, transition state T2 is not important anymore as it is shown by its XTOF value of 0.00. This can be explained in view of eq. (2) that makes the position of a transition state relative to the TDI important. T2 is positioned before the new TDI I5c, whereas it was positioned after the old TDI I1, when spectator species were not taken into account.

As a result, the only influential transition state is T5 (the TDTS). Therefore, the search for a better catalyst has to be based on structures I5c and T5 and might have been misguided by not taking the spectator species into account.

The Influence of Reactants and Products

Until now, we focussed on state energies. The second key factor influencing the efficiency of a given catalyst is the concentration of reactants and/or products. The nature and extent of their influence is of importance, if we wish to couple theory with experiment. Equation (6), which derives from eq. (1), takes into account the concentration effects on the TOF3:

equation image(6)

Here, the numerator Δ′ includes in its first term all the reactant concentrations and in the second term all the product concentrations. Thus, the reaction's driving force increases with higher reactant concentration and decreases with the product concentration. The driving force is exactly zero when the system is in equilibrium, thus, making the TOF equal to zero. This can be shown by adapting the well-known relation between the equilibrium constant K and the Gibbs free energy as shown in eq. (7). On the other hand, the usual exergonic situation with negative ΔGr and high reactant concentrations will provide a positive catalytic current.

equation image(7)
equation image
equation image

In the denominator of the TOF eq. (6), each of the Mi,j terms still consists of a combination of one transition state energy and one intermediate energy as in eq. (1), but now we also multiply by all the δPn,i,j and δRn,i,j values, i.e., all the product concentrations that are generated in the section of the cycle starting with Ij and ending with Ti, and all the reactant concentrations that are consumed outside the Ij to Ti segment. However, for understanding the energetic span model, it is just important to note that the reactant and product concentrations enter into the TOF eq. (6) in the numerator and in the denominator.

To apply eq. (6) to the hydroamination example, we define the concentrations for NH3, C2H4, and C2H5NH2 to be 0.1 mol L−1. This mimics the situation where half the reactants have transformed to product. At this point, an experimental system will certainly have reached a steady state regime, which is one of the assumptions underlying the model. The TOF under these conditions is predicted to be 1.0 h−1, which is slightly lower than the predicted TOF of 1.6 h−1, where no concentrations were taken into account. One can see that the concentrations have a significant influence on the TOF, though their influence is not as great as the influence of the energies (a linear vs. an exponential dependence).

It is important to note that the TOF predicted by the energetic span model is not an average TOF after some conversion, but the current TOF at the specified conditions. For direct comparison, the experimentalist has to measure a concentration vs. time curve. The slope of a tangent at given concentrations corresponds to the TOF estimated by the energetic span model.

The description until this point presents the energetic span model in its current state of development. The next section describes new features and extensions mainly focusing on further concentration effects.

Extensions of the Energetic Span Model

So far the energetic span model takes concentrations into account only for calculating the TOF.3, 4 Herein, we extend the model in the following directions:

  • 1to include the reactant and product concentrations in the degree of TOF control of the energies and
  • 2to determine a degree of TOF control of the reactant and product concentrations, i.e., a quantitative prediction of the influence of the various concentrations on the TOF.

The degrees of TOF control of the reactant and product concentrations can provide explicit predictions regarding the most favorable conditions to be applied in the experiment, e.g., if it helps using a reactant in high concentration, or if a product kinetically inhibits the reaction and should, therefore, be continuously extracted from the reaction mixture.

Modified Expression for the Degree of TOF Control of the Energies

Until now, the degrees of TOF control XTOF [eqs. (3a) and (3b)] were calculated using the Mi,j values as defined by the solely energy based representation of eq. (1), which means the XTOF values did not change when the concentrations of products and reactants were included in the prediction of the TOF. To include the reactant and product concentrations in the degree of TOF control of the energies, we redefine the degree of TOF control to be calculated from the Mi,j terms as defined in eq. (6), where concentrations are taken into account. This leads directly to eqs. (8a) and (8b):

equation image(8a)
equation image(8b)

The concentrations will influence the degree of TOF control significantly, when either several intermediates or transition states are important (i.e., more than one intermediate or more than one TS have an XTOF > 0.00), or when extreme concentrations are involved, e.g., when the solvent acts as reactant or only traces of a reactant are left. Applying this new definition to the hydroamination leads to the XTOF values shown in Table 4. Here, again a concentration of 0.1 mol L−1 was chosen for ammonia, ethylene, and ethylamine.

Table 4. XTOF Values for the Hydroamination Taking Concentrations into Account
IntermediatesTransition states

Comparing the XTOF values calculated with concentration effect (Table 4) to those calculated without concentration effect (Table 3), one can see that under the chosen conditions the influence of I5c is lower than estimated before, whereas the influence of I2b and I5b is increased. Additionally, I1 now has a significant influence, which is even greater than that of I2b and I5b. From this example, we can see the importance of taking concentrations into account also for the XTOF values.

Table 5. The Degrees of TOF Control of the Reactant and Product Concentrations
  ReactantProductXTOF,math imageXTOF,math image
  1. Xmath image = 0.17 + 1.00 − 0.29 = 0.88; ∑Xmath image = 0.29 + 0.28 − 0.92 = −0.35.

Step 1I1I2bNH3 enters 0.17 
Step 2I2bI2C2H4 entersNH3 leaves0.29−0.29
Step 3I2I3    
Step 4I3I4    
Step 5I4I5cC2H4 enters 0.28 
Step 6I5cI5b C2H4 leaves −0.92
Step 7I5bI5NH3 enters 1.00 
Step 8I5I1 C2H5NH2 leaves −0.00

Degree of TOF Control of the Concentrations

Equations (8a) and (8b) reflect the influence of each energy state on the TOF at given values for the concentrations of reactants and products. Yet, it is also desirable to understand the influence of the concentrations themselves on the TOF, as concentrations are easily adjustable parameters within an experimental setup. Using again Campbell's concept,10, 11 but this time applied to the concentrations, we can express the degree of TOF control of the concentrations as shown in eq 9.

equation image(9)

Replacing herein Δ′ and M′, according to eq. (6), we have to distinguish between reactant and product concentrations, as derived in the final eqs. (10a) and (10b):

equation image(10a)
equation image(10b)

The terms equation image and equation image are all the Mi,j terms that contain the kth reactant or product, respectively. These equations can be neatly interpreted within the picture of the thermodynamic driving force Δ′ and the kinetic resistance M′. The first fraction of equation (10a) is the change in the catalytic potential caused by a change in the reactant concentration. The numerator of this term is always positive, whereas the denominator is the catalytic potential itself, which has to be positive as well so that the reaction can proceed forward. Furthermore, the denominator cannot be greater than the numerator, so this term is always ≥1. The second fraction of eq. (10a) accounts for the influence of the reactant concentration on the kinetic resistance for the reaction flux, and it is always <1. Therefore, the difference of the two terms results in a positive XTOF,math image. This means that raising a reactant concentration always raises the TOF, which is consistent with expectations. In eq. (10b), both parts are negatively signed, so XTOF,math image is always negative, meaning that raising a product concentration lowers the TOF.

Applying this concept to the hydramination example leads to the interesting case of NH3 and C2H4 acting not only as reactants for the catalytic cycle but also as reactants and products during the formation and decomposition of the spectator species I2b and I5c. For instance, while going from I2b to I2 ammonia is released as a “product”, Table 5 shows the degrees of TOF control of the reactants and products at the different steps.

The product C2H5NH2 has an XTOF value of 0.00, so the reaction does not suffer from product inhibition. As a rule of thumb, one can say that only those reactants and products that enter or leave the reaction between the TDI and the TDTS are influential.3 However, it gets more complicated when more than one intermediate or more than one transition state have a significant XTOF.

As NH3 and C2H4 act both as reactant and product, their overall influence can only be seen from the sum of their XTOF values. Summing up all the XTOF values for NH3 gives 0.88, whereas C2H4 gives −0.35. This means that raising the concentration of C2H4 would actually lower the TOF. This is consistent with the chemical intuition, as an increased C2H4 concentration would promote the formation of the spectator species I5c and, therefore, the reaction is slowed down. As Table 6 shows, raising the concentration of C2H4 to 0.2 mol L−1 indeed leads to a lowering of the TOF to 0.70 h−1 (entry B). Furthermore, it can be seen that the degrees of TOF control themselves also change with the concentration. A further increase of the C2H4 concentration would lead to an even stronger decrease of the TOF as its degree of TOF control has changed from −0.35 to −0.75. On the other hand, an increase of the NH3 concentration increases the TOF as NH3 has a positive XTOF. Raising its concentration to 0.2 mol L−1 leads to a TOF of 1.9 h−1 (entry C). One can also see that an increased NH3 concentration lowers the inhibition effect of C2H4, as the XTOF value of C2H4 is changed from −0.35 to −0.21. This, too, is in line with chemical intuition, as NH3 competes with C2H4 to drive back the formation of I5c, while at the same time fostering the formation of I5, which continues the catalytic cycle.

Table 6. XTOF and TOF at Different Concentrations
Entryc(NH3) (mol L−1)c(C2H4) (mol L−1)Xmath imageXmath imageTOF (h−1)
  1. The concentration of C2H5NH2 was kept to 0.1 mol L−1.


Optimizing the concentrations leads to a sevenfold increased TOF of 7.1 h−1 at an NH3 concentration of 2.0 mol L−1 and a C2H4 concentration of 0.2 mol L−1 (entry D). Increasing the C2H4 concentration further very soon leads to a negative XTOF again and lowers the TOF (entry E). An additional increase of the NH3 concentration on the other hand would still increase the TOF (entry F). However, this effect has become smaller with the increased NH3 concentration as indicated by the XTOF value for NH3 of 0.37 (entry D) and 0.25 (entry F) compared to 0.88 (entry A) and 0.96 (entry B) at lower concentrations. So, at some point one has to decide if the handling of highly concentrated NH3 is worth the gain in TOF. However, it is clear that ammonia should be used in excess to ethylene in this reaction system.

AUTOF Program for Assessment of Catalytic Cycles

In this section, we describe a program (so-called AUTOF) that was applied to perform all the calculations necessary for the discussion of the example reaction above. AUTOF calculates the degree of TOF control for each state and concentration and the resulting TOF of the cycle according to eqs. (6), (8a), (8b), (10a), and (10b).

The input file must contain all the quantum mechanically computed energies of the intermediates and transition states of the cycle, as well as the concentrations of the reactants and products entering and leaving the cycle, and the temperature (Table 7). If no reactant enters or product leaves, an x has to be entered instead of the respective concentration. Entering an x for every concentration will produce results according to the model in its state before concentrations were taken into account (eqs. 1, 3a, and 3b).

Table 7. Input File Structure for the AUTOF Program
N° of Steps
equation imageequation imageequation imageequation image

The AUTOF program including an executable, the Fortran90 source code, a manual, and an example input file can be obtained free of charge from the authors via email. It requires signing a license agreement that can be obtained from the Supporting Information or by writing to the authors. A flowchart of the program is shown in Figure 4. Examples of input and output of the AUTOF program are also included in the Supporting Information.

Figure 4.

Flowchart of the AUTOF Program.


In this work, we presented an extensive and practical introduction into the energetic span model developed by Kozuch and Shaik1–6 allowing the reader to learn how the model can be applied without going through the detailed mathematical derivations. As an extension to the model, we redefined the degree of TOF control of the intermediate and transition state energies by considering the influence of the concentrations of products and reactants. Furthermore, we introduced the degrees of TOF control of the reactant and product concentrations themselves. This allows the prediction of optimized experimental conditions as was shown by the example of a catalyzed hydroamination reaction of ethylene with ammonia.8 Thereby, we created a link between the theoretically computed catalytic cycle and its experimental counterpart.

We present a program (AUTOF) that makes full use of the complete model and can be used in a black box fashion. However, we recommend to thoroughly read the given examples and explanations to make sure the model is properly applied. The program is available from the authors free of charge. It only requires sending to the authors a signed license agreement included in the Supporting Information. The results provided by the energetic span model and the AUTOF program provide insight into the key factors of a computationally studied catalytic cycle, with the concomitant information of the species that are worth to “tune” experimentally to improve the kinetics of the cycle.